A Sierpinski number is an odd natural number k such that is not prime for all natural numbers n. In 1960, Sierpinski proved that there are infinitely many odd integers k with this property, but failed to give an example. Numbers in such a set with odd k and k < 2^n are called Proth numbers.

Sierpinski Problem

The Sierpinski problem asks what the smallest Sierpinski number is. In 1967, Sierpinski and Selfridge conjectured that 78557 is the smallest Sierpinski number. To show this is true, we need to show that all the odd numbers smaller than 78557 are not Sierpinski numbers, i.e. for every odd k below 78557 there is a positive interger n such that is prime. There are only five numbers which have not been eliminated:

A generalised Fermat Prime is a prime number of the form for a >0. It is called ‘generalised’ as a Fermat Prime is a number of this form with a = 0.

The discovery was made by Sylvanus A. Zimmerman of the United States.

“Until now only 392 generalised Fermat primes had been found: this new discovery makes 393. At 6,253,210 digits long, it’s now the 12th largest of all known primes, and the second-largest known non-Mersenne prime.”

Recently, James Davis found a counterexample to John H. Conway’s ‘Climb to a Prime’ conjecture, for which Conway was offering $1,000 for a solution.

The conjecture states the following:

“Let n be a positive integer. Write the prime factorisation in the usual way, where the primes are written in ascending order and exponents of 1 are omitted. Then bring the exponents down to the line, omit the multiplication signs, giving a number f(n). Now repeat.”

where p is the largest prime factor of n. This motivated him to look for x of the form

The number Davis found was 13532385396179 = 13 x 53^2 x 3853 x 96179, which maps to itself under f (i.e. its a fixed point). So, f will never map this composite number to a prime, hence disproving the conjecture.

The Ulam Spiral, discovered in 1963 by Stanislaw Ulam, is a graphical depiction of the set of prime numbers.

If you were to arrange the positive numbers in a spiral, starting with one at the centre, then circle all of the prime numbers, what would you get? As prime numbers don’t have a predictive structure, you would expect to get little or even nothing out of arranging the primes this way. But, Ulam discovered something incredible:

Ulam Spiral

To his surprise, the circled numbers tended to line up along diagonal lines. In the 200×200 Ulam spiral shown above, diagonal lines are clearly visible, confirming the pattern. Although less prominent, horizontal and vertical lines can also be seen.

Even more amazing, this pattern still appears even if we don’t start with 1 at the centre!

There are many patterns on this plot. One of the simplest ones is that there are many integer constants b and c such that the function:

generates, a number of primes that is large by comparison with the proportion of primes among numbers of similar magnitude, as n counts up {1, 2, 3, …}.

A Leyland number is an integer of the form , where x and y are integers greater than 1. This condition is very important as, without it, every positive integer would be a Leyland number of the form x1 + 1x.

They are named after Paul Leyland, a British number theorist who studied the factorisation of integers and primality testing.

Leyland numbers are of interest as some of them are very large primes.

Leyland Primes

A Leyland prime is a Leyland number that is also prime. The first of such primes are:

Today I thought I’d quickly discuss a extremely important theorem in one of my favourite areas in mathematics: Number Theory (as you can probably tell by the number of posts that I’ve published about primes!).

Perhaps the first property of π(x) – the number of primes less than or equal to x – is that π(x) tends to infinity as x tends to infinity. In other words, the prime numbers are infinite, which was proved by Euclid in “Elements”. A more precise result, established by Euler in 1737, was that the series of reciprocals of the prime numbers:

is a divergent series. In doing so, Euler found an alternative way to prove that there was an infinite number of primes, as if there wasn’t then the series would have a finite value.

The Prime Number Theorem states that if π(x) is the number of primes less than or equal to x, then

Although the notation ~ may be unfamiliar, it simply means that π(x) is asymptotically equal to x/lnx, i.e.

Note that the prime number theorem is equivalent to saying that the nth prime number pn satisfies the following relationship:

The PNT was proposed by Gauss in 1792 when he was only 15 years old! (Makes you wonder what you’ve been doing with your life so far…) He later refined this estimate to